Questions: Part A If the electron is initially in the ground state level of the finite square well, E1=0.625 Einfty, and absorbs a photon, what maximum wavelength can the photon have and still liberate the electron from the finite well? Express your answer numerically in meters using three significant figures.

Part A

If the electron is initially in the ground state level of the finite square well, E1=0.625 Einfty, and absorbs a photon, what maximum wavelength can the photon have and still liberate the electron from the finite well? Express your answer numerically in meters using three significant figures.

Transcript text: Part A If the electron is initially in the ground state level of the finite square well, $E_{1}=0.625 E_{\infty}$, and absorbs a photon, what maximum wavelength can the photon have and still liberate the electron from the finite well? Express your answer numerically in meters using three significant figures. View Available Hint(s) $\square$ Submit Provide Feedback Next＞

Solution

Solution Steps

Step 1: Understand the Problem

The problem involves an electron in a finite square well potential. The electron is initially in the ground state with energy $ E_1 = 0.625 E_{\infty} $, where $ E_{\infty} $ is the energy required to completely free the electron from the well. We need to find the maximum wavelength of a photon that can liberate the electron from the well.

Step 2: Calculate the Energy Required to Liberate the Electron

To liberate the electron, the photon must provide enough energy to overcome the difference between the energy of the electron in the ground state and the energy required to free it, $ E_{\infty} $.

The energy required is: \[ E_{\text{required}} = E_{\infty} - E_1 = E_{\infty} - 0.625 E_{\infty} = 0.375 E_{\infty} \]

Step 3: Relate Energy to Wavelength

The energy of a photon is related to its wavelength by the equation: \[ E = \frac{hc}{\lambda} \] where $ h $ is Planck's constant ($6.626 \times 10^{-34} \, \text{J}\cdot\text{s}$), $ c $ is the speed of light ($3.00 \times 10^8 \, \text{m/s}$), and $ \lambda $ is the wavelength.

Step 4: Solve for the Maximum Wavelength

Rearrange the equation to solve for the wavelength: \[ \lambda = \frac{hc}{E_{\text{required}}} \]

Substitute $ E_{\text{required}} = 0.375 E_{\infty} $ into the equation: \[ \lambda = \frac{hc}{0.375 E_{\infty}} \]

Step 5: Express the Wavelength in Terms of Known Constants

Since $ E_{\infty} $ is the energy required to free the electron, it can be expressed in terms of known constants. However, without specific values for $ E_{\infty} $, we can only express the wavelength in terms of $ E_{\infty} $.